Hamiltonian of a flux qubit-LC oscillator circuit in the deep-strong-coupling regime

TL;DR

This paper derives and analyzes the Hamiltonian of a flux qubit-LC oscillator circuit in the deep-strong-coupling regime, showing its relation to the quantum Rabi model and highlighting gauge-dependent differences.

Contribution

It provides a detailed derivation of the circuit Hamiltonian, compares it with the quantum Rabi Hamiltonian, and discusses gauge transformations and their implications in the deep-strong-coupling regime.

Findings

The circuit Hamiltonian closely matches the quantum Rabi model up to the seventh excited state.

The spectrum shows an overall shift but retains the energy level structure in the deep-strong-coupling regime.

Gauge transformations lead to differences in the Hamiltonian representation that cannot be approximated by simple variants of the quantum Rabi model.

Abstract

We derive the Hamiltonian of a superconducting circuit that comprises a single-Josephson-junction flux qubit inductively coupled to an LC oscillator, and we compare the derived circuit Hamiltonian with the quantum Rabi Hamiltonian, which describes a two-level system coupled to a harmonic oscillator. We show that there is a simple, intuitive correspondence between the circuit Hamiltonian and the quantum Rabi Hamiltonian. While there is an overall shift of the entire spectrum, the energy level structure of the circuit Hamiltonian up to the seventh excited states can still be fitted well by the quantum Rabi Hamiltonian even in the case where the coupling strength is larger than the frequencies of the qubit and the oscillator, i.e., when the qubit-oscillator circuit is in the deep-strong-coupling regime. We also show that although the circuit Hamiltonian can be transformed via a unitary transformation to a Hamiltonian containing a capacitive coupling term, the resulting circuit Hamiltonian cannot be approximated by the variant of the quantum Rabi Hamiltonian that is obtained using an analogous procedure for mapping the circuit variables onto Pauli and harmonic oscillator operators, even for relatively weak coupling. This difference between the flux and charge gauges follows from the properties of the qubit Hamiltonian eigenstates.